The immune system plays a dual role during neoplastic progression. It can suppress tumor growth by eliminating cancer cells, and also promote neoplastic expansion by either selecting for tumor cells that are fitter to survive in an immunocompetent host or by establishing the right conditions within the tumor microenvironment. First, I present a model to study the dynamics of subclonal evolution of cancer. I model selection through time as an epistatic process. That is, the fitness change in a given cell is not simply additive, but depends on previous mutations. Simulation studies indicate that tumors are composed of myriads of small subclones at the time of diagnosis. Because some of these rare subclones harbor pre-existing treatment-resistant mutations, they present a major challenge to precision medicine. Second, I study the question of self and non-self discrimination by the immune system, which is fundamental in the field in cancer immunology. By performing a quantitative analysis of the biochemical properties of thousands of MHC class I peptides, I find that hydrophobicity of T cell receptors contact residues is a hallmark of immunogenic epitopes. Based on these findings, I further develop a computational model to predict immunogenic epitopes which facilitate the development of T cell vaccines against pathogen and tumor antigens. Lastly, I study the effect of early detection in the context of Ebola. I develope a simple mathematical model calibrated to the transmission dynamics of Ebola virus in West Africa. My findings suggest that a strategy that focuses on early diagnosis of high-risk individuals, caregivers, and health-care workers at the pre-symptomatic stage, when combined with public health measures to improve the speed and efficacy of isolation of infectious individuals, can lead to rapid reductions in Ebola transmission.

In the honey bee antennal lobe, uniglomerular projection neurons (uPNs) transiently spike to odor sensory stimuli with odor-specific response latencies, i.e., delays to first spike after odor

stimulation onset. Recent calcium imaging studies show that the spatio-temporal response profile of the activated uPNs are dynamic and changes as a result

of associative conditioning, facilitating odor-detection of learned odors.

Moreover, odor-representation in the antennal lobe undergo reward-mediated plasticity processes that increase response delay variations

in the activated ensemble of uniglomerular projection neurons. Octopamine is necessarily involved in these plasticity processes. Yet, the cellular mechanisms are not

well understood. I hypothesize that octopamine modulates cholinergic transmission to uPNs by triggering translation

and upregulation of nicotinic receptors, which are more permeable to calcium. Consequently, this increased calcium-influx signals transcription factors that upregulate potassium

channels in the dendritic cortex of glomeruli, similar to synaptic plasticity mechanisms recently

shown in various insect species. A biophysical model of the antennal lobe circuit is developed in order to test the hypothesis that increased potassium channel expression in uPNs mediate response delays to first

spike, dynamically tuning odor-representations to facilitate odor-detection of learned odors.

Combination therapy has shown to improve success for cancer treatment. Oncolytic virotherapy is cancer treatment that uses engineered viruses to specifically infect and kill cancer cells, without harming healthy cells. Immunotherapy boosts the body's natural defenses towards cancer. The combination of oncolytic virotherapy and immunotherapy is explored through deterministic systems of nonlinear differential equations, constructed to match experimental data for murine melanoma. Mathematical analysis was done in order to gain insight on the relationship between cancer, viruses and immune response. One extension of the model focuses on clinical needs, with the underlying goal to seek optimal treatment regimens; for both frequency and dose quantity. The models in this work were first used to estimate parameters from preclinical experimental data, to identify biologically realistic parameter values. Insight gained from the mathematical analysis in the first model, allowed for numerical analysis to explore optimal treatment regimens of combination oncolytic virotherapy and dendritic vaccinations. Permutations accounting for treatment scheduled were done to find regimens that reduce tumor size. Observations from the produced data lead to in silico exploration of immune-viral interactions. Results suggest under optimal settings, combination treatment works better than monotherapy of either type. The most optimal result suggests treatment over a longer period of time, with fractioned doses, while reducing the total dendritic vaccination quantity, and maintaining the maximum virotherapy used in the experimental work.

According to the Centers for Disease Control and Prevention (CDC), type 2 diabetes accounts for 90-95% of diabetes (29.1 million) cases and manifests in 15-30% of prediabetes (86 million) cases, where 9 out of 10 individuals do not know they have prediabetes. Obesity, observed in 56.9% of diabetes cases, arises from the interactions among genetic, biological, environmental, and behavioral factors that are not well understood. Assessing the strength of these links in conjunction with the identification and evaluation of intervention strategies in vulnerable populations is central to the study of chronic diseases. This research addresses three issues that loosely connect three levels of organization utilizing a combination of quantitative and qualitative methods. First, the nonlinear dynamics between insulin, glucose, and free fatty acids is studied via a hypothesis-based model and validated with bariatric surgery data, demonstrating key metabolic factors for maintaining glucose homeostasis. Second, the challenges associated with the treatment or management, and prevention of diabetes is explored in the context of an individualized-based intervention study, highlighting the importance of diet and environment. Third, the importance of tailored school lunch programs and policies is studied through contagion models developed within a social ecological framework. The Ratatouille Effect, motivated by a pilot study among PreK-8th grade Arizona students, is studied and exposes the importance of institutionalizing practical methods that factor in the culture, norms, and values of the community. The outcomes of this research illustrate an integrative framework that bridges physiological, individual, and population level approaches to study type 2 diabetes and obesity from a holistic perspective. This work reveals the significance of utilizing quantitative and qualitative methods to better elucidate underlying causes of chronic diseases and for developing solutions that lead to sustainable healthy behaviors, and more importantly, the need for translatable multilevel methodologies for the study of the progression, treatment, and prevention of chronic diseases from a multidisciplinary perspective.

The Visceral Leishmaniasis (VL) is primarily endemic in five countries, with India and Sudan having the highest burden. The risk factors associated with VL are either unknown in some regions or vary drastically among empirical studies. Here, a dynamical model, motivated and informed by field data from the literature, is analyzed and employed to identify and quantify the impact of region dependent risks on the VL transmission dynamics. Parameter estimation procedures were developed using model-derived quantities and empirical data from multiple resources. The dynamics of VL depend on the estimates of the control reproductive number, RC, interpreted as the average number of secondary infections generated by a single infectious individual during the infectious period. The distribution of RC was estimated for both India (with mean 2.1 ± 1.1) and Sudan (with mean 1.45 ± 0.57). This suggests that VL can be established in naive regions of India more easily than in naive regions of Sudan. The parameter sensitivity analysis on RC suggests that the average biting rate and transmission probabilities between host and vector are among the most sensitive parameters for both countries. The comparative assessment of VL transmission dynamics in both India and Sudan was carried out by parameter sensitivity analysis on VL-related prevalences (such as prevalences of asymptomatic hosts, symptomatic hosts, and infected vectors). The results identify that the treatment and symptoms’ developmental rates are parameters that are highly sensitive to VL symptomatic and asymptomatic host prevalence, respectively, for both countries. It is found that the estimates of transmission probability are significantly different between India (from human to sandflies with mean of 0.39 ± 0.12; from sandflies to human with mean 0.0005 ± 0.0002) and Sudan (from human to sandflies with mean 0.26 ± 0.07; from sandflies to human with mean 0.0002 ± 0.0001). The results have significant implications for elimination. An increasing focus on elimination requires a review of priorities within the VL control agenda. The development of systematic implementation of con­trol programs based on identified risk factors (such as monitoring of asymptomatically infected individuals) has a high transmission-blocking potential.

The 2009-10 influenza and the 2014-15 Ebola pandemics brought once again urgency to an old question: What are the limits on prediction and what can be proposed that is useful in the face of an epidemic outbreak?

This thesis looks first at the impact that limited access to vaccine stockpiles may have on a single influenza outbreak. The purpose is to highlight the challenges faced by populations embedded in inadequate health systems and to identify and assess ways of ameliorating the impact of resource limitations on public health policy.

Age-specific per capita constraint rates play an important role on the dynamics of communicable diseases and, influenza is, of course, no exception. Yet the challenges associated with estimating age-specific contact rates have not been decisively met. And so, this thesis attempts to connect contact theory with age-specific contact data in the context of influenza outbreaks in practical ways. In mathematical epidemiology, proportionate mixing is used as the preferred theoretical mixing structure and so, the frame of discussion of this dissertation follows this specific theoretical framework. The questions that drive this dissertation, in the context of influenza dynamics, proportionate mixing, and control, are:

I. What is the role of age-aggregation on the dynamics of a single outbreak? Or simply speaking, does the number and length of the age-classes used to model a population make a significant difference on quantitative predictions?

II. What would the age-specific optimal influenza vaccination policies be? Or, what are the age-specific vaccination policies needed to control an outbreak in the presence of limited or unlimited vaccine stockpiles?

Intertwined with the above questions are issues of resilience and uncertainty including, whether or not data collected on mixing (by social scientists) can be used effectively to address both questions in the context of influenza and proportionate mixing. The objective is to provide answers to these questions by assessing the role of aggregation (number and length of age classes) and model robustness (does the aggregation scheme selected makes a difference on influenza dynamics and control) via comparisons between purely data-driven model and proportionate mixing models.

Statistical Methods have been widely used in understanding factors for clinical and public health data. Statistical hypotheses are procedures for testing pre-stated hypotheses. The development and properties of these procedures as well as their performance are based upon certain assumptions. Desirable properties of statistical tests are to maintain validity and to perform well even if these assumptions are not met. A statistical test that maintains such desirable properties is called robust. Mathematical models are typically mechanistic framework, used to study dynamic interactions between components (mechanisms) of a system, and how these interactions give rise to the changes in behavior (patterns) of the system as a whole over time.

In this thesis, I have developed a study that uses novel techniques to link robust statistical tests and mathematical modeling methods guided by limited data from developed and developing regions in order to address pressing clinical and epidemiological questions of interest. The procedure in this study consists of three primary steps, namely, data collection, uncertainty quantification in data, and linking dynamic model to collected data.

The first part of the study focuses on designing, collecting, and summarizing empirical data from the only national survey of hospitals ever conducted regarding patient controlled analgesia (PCA) practices among 168 hospitals across 40 states, in order to assess risks before putting patients on PCA. I used statistical relational models and exploratory data analysis to address the question. Risk factors assessed indicate a great concern for the safety of patients from one healthcare institution to other.

In the second part, I quantify uncertainty associated with data obtained from James A Lovell Federal Healthcare Center to primarily study the effect of Benign Prostatic Hypertrophy (BPH) on sleep architecture in patients with Obstructive Sleep Apnea (OSA). Patients with OSA and BPH demonstrated significant difference in their sleep architecture in comparison to patients without BPH. One of the ways to validate these differences in sleep architecture between the two groups may be to carry out a similar study that evaluates the effect of some other chronic disease on sleep architecture in patients with OSA.

Additionally, I also address theoretical statistical questions such as (1) how to estimate the distribution of a variable in order to retest null hypothesis when the sample size is limited, and (2) how changes on assumptions (like monotonicity and nonlinearity) translate into the effect of the independent variable on the outcome variable. To address these questions we use multiple techniques such as Partial Rank Correlation Coefficients (PRCC) based sensitivity analysis, Fractional Polynomials, and statistical relational models.

In the third part, my goal was to identify socio-economic-environment-related risk factors for Visceral Leishmaniasis (VL) and use the identified critical factors to develop a mathematical model to understand VL transmission dynamics when data is highly underreported. I primarily studied the role of age-specific- susceptibility and epidemiological quantities on the dynamics of VL in the Indian state of Bihar. Statistical results provided ideas on the choice of the modeling framework and estimates of model parameters.

In the conclusion, this study addressed three primary theoretical modeling-related questions (1) how to analyze collected data when sample size limited, and how modeling assumptions varies results of data analysis? (2) Is it possible to identify hidden associations and nonlinearity of these associations using such underpowered data and (3) how statistical models provide more reasonable structure to mathematical modeling framework that can be used in turn to understand dynamics of the system.

The Mathematical and Theoretical Biology Institute (MTBI) is a summer research program for undergraduate students, largely from underrepresented minority groups. Founded in 1996, it serves as a 'life-long' mentorship program, providing continuous support for its students and alumni. This study investigates how MTBI supports student development in applied mathematical research. This includes identifying of motivational factors to pursue and develop capacity to complete higher education.

The theoretical lens of developmental psychologists Lev Vygotsky (1978, 1987) and Lois Holzman (2010) that sees learning and development as a social process is used. From this view student development in MTBI is attributed to the collaborative and creative way students co-create the process of becoming scientists. This results in building a continuing network of academic and professional relationships among peers and mentors, in which around three quarters of MTBI PhD graduates come from underrepresented groups.

The extent to which MTBI creates a Vygotskian learning environment is explored from the perspectives of participants who earned doctoral degrees. Previously hypothesized factors (Castillo-Garsow, Castillo-Chavez and Woodley, 2013) that affect participants’ educational and professional development are expanded on.

Factors identified by participants are a passion for the mathematical sciences; desire to grow; enriching collaborative and peer-like interactions; and discovering career options. The self-recognition that they had the ability to be successful, key element of the Vygotskian-Holzman theoretical framework, was a commonly identified theme for their educational development and professional growth.

Participants characterize the collaborative and creative aspects of MTBI. They reported that collaborative dynamics with peers were strengthened as they co-created a learning environment that facilitated and accelerated their understanding of the mathematics needed to address their research. The dynamics of collaboration allowed them to complete complex homework assignments, and helped them formulate and complete their projects. Participants identified the creative environments of their research projects as where creativity emerged in the dynamics of the program.

These data-driven findings characterize for the first time a summer program in the mathematical sciences as a Vygotskian-Holzman environment, that is, a `place’ where participants are seen as capable applied mathematicians, where the dynamics of collaboration and creativity are fundamental components.

Mathematical epidemiology, one of the oldest and richest areas in mathematical biology, has significantly enhanced our understanding of how pathogens emerge, evolve, and spread. Classical epidemiological models, the standard for predicting and managing the spread of infectious disease, assume that contacts between susceptible and infectious individuals depend on their relative frequency in the population. The behavioral factors that underpin contact rates are not generally addressed. There is, however, an emerging a class of models that addresses the feedbacks between infectious disease dynamics and the behavioral decisions driving host contact. Referred to as “economic epidemiology” or “epidemiological economics,” the approach explores the determinants of decisions about the number and type of contacts made by individuals, using insights and methods from economics. We show how the approach has the potential both to improve predictions of the course of infectious disease, and to support development of novel approaches to infectious disease management.

Analysis of social networks has the potential to provide insights into wide range of applications. As datasets continue to grow, a key challenge is the lack of a widely applicable algorithmic framework for detection of statistically anomalous networks and network properties. Unlike traditional signal processing, where models of truth or empirical verification and background data exist and are often well defined, these features are commonly lacking in social and other networks. Here, a novel algorithmic framework for statistical signal processing for graphs is presented. The framework is based on the analysis of spectral properties of the residuals matrix. The framework is applied to the detection of innovation patterns in publication networks, leveraging well-studied empirical knowledge from the history of science. Both the framework itself and the application constitute novel contributions, while advancing algorithmic and mathematical techniques for graph-based data and understanding of the patterns of emergence of novel scientific research. Results indicate the efficacy of the approach and highlight a number of fruitful future directions.